Viral processes by random walks on random regular graphs

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^ε$, where $n$ is the number of vertices in the random graph, and $ε$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős-Rényi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $β\in(0,1)$ determined by the parameters of the model, $βk$ get infected with probability $β$, and $O(\ln k)$ get infected with probability $(1-β)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.

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